The sparsity pattern of the estimator \(\widehat{S}\)  can also be justified on the grounds of
conditional independence properties of \(\pi \) —the so-called Markov properties.
These properties can be collected in a simple undirected graph
\(\mathcal {G}= (\mathcal {V}, \mathcal {E})\) , where each node \(i \in \mathcal {V}\)  is associated with one of the random variables in the collection \(Z=(Z_1,\ldots ,Z_n)\) , and where
each
edge \((i,j) \in \mathcal {E}\)  encodes a notion of probabilistic interaction between the pair \(Z_i,Z_j\) .
We say that \(Z\)  (or \(\pi \) ) is a Markov random field (MRF) with respect to \(\mathcal {G}\)  if for any partition \(\mathcal {A},\mathcal {S},\mathcal {B}\)  of the vertex set—for which
\(\mathcal {S}\)  is a separator set\(\mathcal {S}\)  is a separator set for \(\mathcal {A}\)  and \(\mathcal {B}\)  if every path from a node in
\(\mathcal {A}\)  to a node in \(\mathcal {B}\)  goes through \(\mathcal {S}\) .
for \(\mathcal {A}\)  and \(\mathcal {B}\) —we have that
\(Z_{\mathcal {A}}\)  and \(Z_{\mathcal {B}}\)  are conditionally independent given \(Z_{\mathcal {S}}\) ,
i.e., \(Z_{\mathcal {A}}  \perp \!\!\!\perp Z_{\mathcal {B}} \vert Z_{\mathcal {S}}\)  [1]}.
See Figure REF  (left)
for an example.
For any strictly positive \(\pi \) , we can find
the sparsest
\(\mathcal {G}\)  that is compatible with the conditional independence structure of \(\pi \)  as follows:
let
\((i,j)\notin \mathcal {E}\)  if and only if
\(Z_{i}  \perp \!\!\!\perp Z_{j} \vert Z_{\mathcal {V}\setminus \lbrace i,j\rbrace }\) 
[2]}.
